I would avoid drilling an SA-22 stock at all cost. Several manufacturers make 'no swivel' slings (mostly for shotguns) that will work fine for a Browning .22 auto. For example: https://www.butlercreek.com/slings/nylon-shotgun-sling
Cabelas and Bass Pro probably carry something similar. They rely on velcro fastening around the grip and barrel. It sounds weak, but Velcro is very strong in shear. On the SA-22, there is simply too little wood to safely mount a front swivel, IMO. I would expect the velcro to be at least as strong.
I think the one I picked up a few years ago at a LGS was made by Uncle Mike's.
I've never carried my SA-22 using this sling, but wouldn't hesitate.